energy-momentum conservation

[dpasquie]

Dear all,

I am wondering if there are any constraints on the electron-phonon matrix elements coming from energy conservation, something like g_{n,m}(k,q)=0 unless \epsilon_{m,k+q}-\epsilon_{n,k}=\omega_q. It seems that it is not the case because otherwise quantities like the electron-phonon coupling strength would be zero. Am I missing someting?

If this is not the case, how do the matrix elements decay as a function of the difference of energy of the electrons? I am asking those questions because I would like to understand if the electron and phonon self-energies receive contributions from states far from the fermi level, as the factor 1/(\epsilon_k-\epsilon_k+q) alone would give rise to rather strong contributions.

I hope my questions are clear enough, thanks in advance!

Diego

[sponce]

Dear Diego,

There are indeed not such condition impose on the electron-phonon matrix elements.

However you are correct that this is imposed on physical quantities.

For example the electron-phonon self-energy Sigma does have such constrain in its form.

Indeed, you can show that the imaginary part of Sigma has \delta(\epsilon_k-\epsilon_(k+q)+\omega_q) terms.

Best,

Samuel

[dpasquie]

Dear Samuel,

Thank you very much for your answering my question.

Best,
Diego